Global properties of vector fields on compact Lie groups in Komatsu classes. II. Normal forms

TL;DR

This paper characterizes the global hypoellipticity and solvability of certain vector field operators on compact Lie groups within Komatsu ultradifferentiable classes, providing conjugation techniques and examples in specific manifolds.

Contribution

It offers a complete characterization of global hypoellipticity and solvability for operators on compact Lie groups in Komatsu classes, including conjugation methods and explicit examples.

Findings

Characterization of global hypoellipticity and solvability in Komatsu classes.

Construction of conjugation between variable and constant coefficient operators.

Examples of operators that are solvable in Gevrey but not in smooth category.

Abstract

Let $G_1$ and $G_2$ be compact Lie groups, $X_1 \in \mathfrak{g}_1$, $X_2 \in \mathfrak{g}_2$ and consider the operator \begin{equation*} L_{aq} = X_1 + a(x_1)X_2 + q(x_1,x_2), \end{equation*} where $a$ and $q$ are ultradifferentiable functions in the sense of Komatsu, and $a$ is real-valued. We characterize completely the global hypoellipticity and the global solvability of $L_{aq}$ in the sense of Komatsu. For this, we present a conjugation between $L_{aq}$ and a constant-coefficient operator that preserves these global properties in Komatsu classes. We also present examples of globally hypoelliptic and globally solvable operators on $\mathbb{T}^1\times \mathbb{S}^3$ and $\mathbb{S}^3\times \mathbb{S}^3$ in the sense of Komatsu. In particular, we give examples of differential operators which are not globally $C^\infty$-solvable, but are globally solvable in Gevrey spaces.